Chevakinsky was born into a noble family in the village of Veshki in the Novotorzhsky Uyezd of Tver Governorate.
In 1729 he entered the Naval Academy in Saint Petersburg, from whence he was assigned to the Izmaylovsky Life Guards Regiment in 1734.
At the request of the Admiralty Board he was discharged for unauthorized absence from the Academy and apprenticed to the architectural company of Ivan Korobov [ru], under whose direction he worked for seven years.
Here Chevakinsky erected two buildings (a church and a hall) connected by galleries to the central part of the palace, erected the Monbizhu pavilion [ru] (which housed palace officials), and participated in the creation of the Hermitage pavilion [ru].
Chevakinsky's largest building in Saint Petersburg was the St. Nicholas Naval Cathedral (1753–1762) with a separate tiered bell tower (1756–1758).